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Math 342 Homework Assignment #3 (Due Thursday, March 3, in class) All numbered problems are worth the same value. 1. a. Verify that 727 is prime. b. Find the multiplicative inverse of 726 in the field Z727 . c. Find the multiplicative inverse of 373 in the field Z727 . 2. Recall the definition of the field GF (4), given by the addition and multiplication tables in the first paragraph of page 36 in the text. Compute the following in this field. (a) a + a (b) (a + 1)(a + b) (c) a1000 3. (a) Show that in a field F , for all a, b, c ∈ F such that a 6= 0, there is a unique solution to the equation ax + b = c. (b) Find a counter-example to part a) if F is merely a commutative ring with identity. (c) Find all solutions to the equation x2 + x + 1 = 0 in the fields GF (2), GF (3) and GF (4). 4. Prove that if p is prime and a ∈ Zp and a 6= 0, then ap−1 = 1 mod p. Hint: Recall from the proof that Zp is a field: {a, 2a, . . . (p − 1)a} are distinct and non-zero mod p; thus, {a, 2a, . . . (p − 1)a} = {1, 2, . . . (p − 1)} ( mod p) and so the following products are equal mod p: (a)(2a)(3a) · · · ((p − 1)a) = (1)(2) · · · (p − 1) ( mod p) 5. Without using a computer or calculator, find the following principal remainders. a. (99)101 mod 31 b. (263912)20111 mod 13 6. Let V be a vector space and u ∈ V . (a) Show that 0u = 0 (b) Show that (−1)u = −u. 7. Which of the following subsets are subspaces of GF (q)n ? (a) {0000, 1111, 1010, 0101}, q = 2, n = 4 (b) {0000, 1111, 1010, 0101}, q = 3, n = 4 1 (c) {000, 111, 222}, q = 3, n = 3 (d) {00, 11, aa, ab}, q = 4, n = 2 (e) {a1ab, bab1, 1b1a}, q = 4, n = 4 8. For each of the following subsets S of GF (q)n , find a basis for, and the dimension of, the span of S. Also, determine if the given S is linearly independent. (a) S = {1100, 1010, 1001, 0101}, q = 2, n = 4 (b) S = {1234, 3142, 2413, 4321}, q = 5, n = 4 (c) S = {0140, 4322, 1233, 2141}, q = 5, n = 4 (d) S = {a1ab, bab1, 1b1a}, q = 4, n = 4 (e) S = {01ab, 1ab0, ab10, abbb}, q = 4, n = 4 2